Theorem suppmptcfin 42160

Description: The support of a mapping with value 0 except of one is finite. (Contributed by AV, 27-Apr-2019.)

Hypotheses

Ref	Expression
suppmptcfin.b	|- B = ( Base ` M )
suppmptcfin.r	|- R = (Scalar ` M )
suppmptcfin.0	|- .0. = ( 0g ` R )
suppmptcfin.1	|- .1. = ( 1r ` R )
suppmptcfin.f	|- F = ( x e. V |-> if ( x = X , .1. , .0. ) )

Assertion

Ref	Expression
suppmptcfin	|- ( ( M e. LMod /\ V e. ~P B /\ X e. V ) -> ( F supp .0. ) e. Fin )

Distinct variable groups:   x, B   x, F   x, M   x, V   x, X   x, .1.   x, .0.

Allowed substitution hint:   R( x)

Proof of Theorem suppmptcfin

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		suppmptcfin.f	. . . 4 |- F = ( x e. V |-> if ( x = X , .1. , .0. ) )
2		eqeq1 2626	. . . . . 6 |- ( x = v -> ( x = X <-> v = X ) )
3	2	ifbid 4108	. . . . 5 |- ( x = v -> if ( x = X , .1. , .0. ) = if ( v = X , .1. , .0. ) )
4	3	cbvmptv 4750	. . . 4 |- ( x e. V |-> if ( x = X , .1. , .0. ) ) = ( v e. V |-> if ( v = X , .1. , .0. ) )
5	1, 4	eqtri 2644	. . 3 |- F = ( v e. V |-> if ( v = X , .1. , .0. ) )
6		simp2 1062	. . 3 |- ( ( M e. LMod /\ V e. ~P B /\ X e. V ) -> V e. ~P B )
7		suppmptcfin.0	. . . . 5 |- .0. = ( 0g ` R )
8		fvex 6201	. . . . 5 |- ( 0g ` R ) e. _V
9	7, 8	eqeltri 2697	. . . 4 |- .0. e. _V
10	9	a1i 11	. . 3 |- ( ( M e. LMod /\ V e. ~P B /\ X e. V ) -> .0. e. _V )
11		suppmptcfin.1	. . . . . 6 |- .1. = ( 1r ` R )
12		fvex 6201	. . . . . 6 |- ( 1r ` R ) e. _V
13	11, 12	eqeltri 2697	. . . . 5 |- .1. e. _V
14	13	a1i 11	. . . 4 |- ( ( ( M e. LMod /\ V e. ~P B /\ X e. V ) /\ v e. V ) -> .1. e. _V )
15	9	a1i 11	. . . 4 |- ( ( ( M e. LMod /\ V e. ~P B /\ X e. V ) /\ v e. V ) -> .0. e. _V )
16	14, 15	ifcld 4131	. . 3 |- ( ( ( M e. LMod /\ V e. ~P B /\ X e. V ) /\ v e. V ) -> if ( v = X , .1. , .0. ) e. _V )
17	5, 6, 10, 16	mptsuppd 7318	. 2 |- ( ( M e. LMod /\ V e. ~P B /\ X e. V ) -> ( F supp .0. ) = { v e. V | if ( v = X , .1. , .0. ) =/= .0. } )
18		snfi 8038	. . 3 |- { X } e. Fin
19		2a1 28	. . . . . 6 |- ( v = X -> ( ( ( M e. LMod /\ V e. ~P B /\ X e. V ) /\ v e. V ) -> ( if ( v = X , .1. , .0. ) =/= .0. -> v = X ) ) )
20		iffalse 4095	. . . . . . . . . 10 |- ( -. v = X -> if ( v = X , .1. , .0. ) = .0. )
21	20	adantr 481	. . . . . . . . 9 |- ( ( -. v = X /\ ( ( M e. LMod /\ V e. ~P B /\ X e. V ) /\ v e. V ) ) -> if ( v = X , .1. , .0. ) = .0. )
22	21	neeq1d 2853	. . . . . . . 8 |- ( ( -. v = X /\ ( ( M e. LMod /\ V e. ~P B /\ X e. V ) /\ v e. V ) ) -> ( if ( v = X , .1. , .0. ) =/= .0. <-> .0. =/= .0. ) )
23		eqid 2622	. . . . . . . . 9 |- .0. = .0.
24		eqneqall 2805	. . . . . . . . 9 |- ( .0. = .0. -> ( .0. =/= .0. -> v = X ) )
25	23, 24	ax-mp 5	. . . . . . . 8 |- ( .0. =/= .0. -> v = X )
26	22, 25	syl6bi 243	. . . . . . 7 |- ( ( -. v = X /\ ( ( M e. LMod /\ V e. ~P B /\ X e. V ) /\ v e. V ) ) -> ( if ( v = X , .1. , .0. ) =/= .0. -> v = X ) )
27	26	ex 450	. . . . . 6 |- ( -. v = X -> ( ( ( M e. LMod /\ V e. ~P B /\ X e. V ) /\ v e. V ) -> ( if ( v = X , .1. , .0. ) =/= .0. -> v = X ) ) )
28	19, 27	pm2.61i 176	. . . . 5 |- ( ( ( M e. LMod /\ V e. ~P B /\ X e. V ) /\ v e. V ) -> ( if ( v = X , .1. , .0. ) =/= .0. -> v = X ) )
29	28	ralrimiva 2966	. . . 4 |- ( ( M e. LMod /\ V e. ~P B /\ X e. V ) -> A. v e. V ( if ( v = X , .1. , .0. ) =/= .0. -> v = X ) )
30		rabsssn 4215	. . . 4 |- ( { v e. V | if ( v = X , .1. , .0. ) =/= .0. } C_ { X } <-> A. v e. V ( if ( v = X , .1. , .0. ) =/= .0. -> v = X ) )
31	29, 30	sylibr 224	. . 3 |- ( ( M e. LMod /\ V e. ~P B /\ X e. V ) -> { v e. V | if ( v = X , .1. , .0. ) =/= .0. } C_ { X } )
32		ssfi 8180	. . 3 |- ( ( { X } e. Fin /\ { v e. V | if ( v = X , .1. , .0. ) =/= .0. } C_ { X } ) -> { v e. V | if ( v = X , .1. , .0. ) =/= .0. } e. Fin )
33	18, 31, 32	sylancr 695	. 2 |- ( ( M e. LMod /\ V e. ~P B /\ X e. V ) -> { v e. V | if ( v = X , .1. , .0. ) =/= .0. } e. Fin )
34	17, 33	eqeltrd 2701	1 |- ( ( M e. LMod /\ V e. ~P B /\ X e. V ) -> ( F supp .0. ) e. Fin )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   ifcif 4086   ~Pcpw 4158   {csn 4177   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   supp csupp 7295   Fincfn 7955   Basecbs 15857  Scalarcsca 15944   0gc0g 16100   1rcur 18501   LModclmod 18863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-supp 7296  df-1o 7560  df-er 7742  df-en 7956  df-fin 7959

This theorem is referenced by:  mptcfsupp  42161